Questions tagged [hopf-algebras]
For questions about Hopf algebras and related concepts, such as quantum groups. The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science.
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Affine group schemes
I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups.
My question
Let $k$ be an ...
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What are the primitive elements of tensor algebra
In a bialgebra $(H,m,u,\Delta,\epsilon)$, subspace of primitive elements are $P(H)=\{x\in H:\Delta (x)=x\otimes 1+1\otimes x\}$. I know that if $x,y$ are primitive, then $[x,y]=xy-yx$ is also ...
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Grouplike Hopf algebras are group rings?
Let $H$ be a commutative and cocommutative Hopf algebra over an algebraically closed field $k$. I've read that if $H$ is grouplike in the sense that it has no nonzero primitive elements, then $H$ is ...
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Antipode of a Hopf algebra being an antihomomorphism: unable to follow the proof
A PhD thesis contains the following proof that antipode of a Hopf algebra is algebra antihomomorphism (page 22):
Here $\nu = \eta \circ \varepsilon$, where $\eta$ is the unit map and $\varepsilon$ is ...
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Solution of the Yang-Baxter equation not coming from quasi-triangular structure
Let $A$ be an associative, unital algebra over a field $\Bbbk$, and let $R \in A \otimes A$ be an invertible element which is a solution of the Yang-Bater equation in $A \otimes A \otimes A$ $$R_{12}...
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Hopf algebra structure on the ring of functions from an infinite group
A basic algebra statement confuses me. This is in Greenlees' Equivariant Formal Group Laws and Complex Oriented Cohomology Theories, p. 229 (in the journal).
If $k$ is a field, $A$ is an infinite ...
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Group isomorphism $\operatorname{Tgt}_e(G) \cong \operatorname{Hom}_{k-\text{linear}}(\ker(\epsilon)/\ker(\epsilon)^2, k)$ for algebraic groups.
I'm reading from Milne's text https://www.jmilne.org/math/CourseNotes/RG.pdf, in chapter 8 on Lie algebras of (affine) algebraic groups $G$ over $k$. In it, he claims in 8.6 that there is an ...
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Understanding a proof of a lemma for rigid categories [closed]
I'm reading the proof of the lemma 3.4 in the Bruguieres' paper on Hopf monads which claims the following:
Lemma Let $F,G: \mathcal{C} \rightarrow \mathcal{D}$ be two strong monoidal functors and $\...
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Commutative diagrams in the definition of bialgebras, what do they mean?
I am reading the definition of Bialgebras over a field $\mathbb{K}$. The definition is the following:
A bialgebra over a field $\mathbb{K}$ is a vector space $B$ over $\mathbb{K}$ equipped with $\...
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About the regular representation of weak hopf algebra
In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now ...
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Non unital Hopf relation
The following problem is an exercise in Loday-Vallette's Algebraic Operads. I hope I am understanding this correctly. Any suggestions or hints would be appreciated.
Show that the restriction of the
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Haar integral of a finite dimensional Hopf algebra: an explicit expression
Let $\mathcal{H}$ be a finite dimensional Hopf algebra. A nonzero element $\Omega\in \mathcal{H}$ is called an integral in $\mathcal{H}$ if $$x~\Omega=\epsilon(x)\Omega,~~\forall x\in \mathcal{H}.\tag{...
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Intuition for Coproduct of Grouplike vs Primitive elements in a Coalgebra?
I'm trying to understand Hopf Algebras as a physicist with a limited background in abstract algebra, and I might be in a little over my head.
In particular I'm trying to wrap my head around the fact ...
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Are antipodes of free, finite rank Hopf algebras over general rings invertible?
It is a well-known result by Larson and Sweedler that, for finite-dimensional Hopf algebras over a field, the antipode is always a linear isomorphism.
My question is whether this property still holds ...
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Definition of equivalent representation of Hopf algebras
I have a question about equivalent representation of Hopf algebras because I am not unfamiliar with Hopf algebras.
Here is my question:
If $\rho_1$ and $\rho_2$ are two representations of a Hopf ...
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